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Let G be a graph on n vertices with maximum degree D. We use the Lov\'aszlocal lemma to show the following two results about colourings c of the edgesof the complete graph K_n. If for each vertex v of K_n the colouring c assignseach colour to at most (n-2)/22.4D^2 edges emanating from v, then there is acopy of G in K_n which is properly edge-coloured by c. This improves on aresult of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4),409-433, 2003]. On the other hand, if c assigns each colour to at most n/51D^2edges of K_n, then there is a copy of G in K_n such that each edge of Greceives a different colour from c. This proves a conjecture of Frieze andKrivelevich [Electron. J. Comb. 15(1), R59, 2008]. Our proofs rely on aframework developed by Lu and Sz\'ekely [Electron. J. Comb. 14(1), R63, 2007]for applying the local lemma to random injections. In order to improve theconstants in our results we use a version of the local lemma due to Bissacot,Fern\'andez, Procacci, and Scoppola [preprint, arXiv:0910.1824].

Properly coloured copies and rainbow copies of large graphs with small maximum degree